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arXiv:1203.0374 · Physical Review A

Implementation of a three-qubit Toffoli gate in a single step

Ai Min Chen, Sam Young Cho, Mun Dae Kim

Published Mar 20, 2012• 13 citations
arXivHTMLPDF

Abstract

Single-step implementations of multiqubit gates are generally believed to provide a simpler design, a faster operation, and a lower decoherence. For three coupled qubits interacting with a photon field, a realizable scheme for a single-step Toffoli gate is investigated. We find that the three-qubit system can be described by four effective modified Jaynes-Cummings models in the states of two control qubits. Within the rotating-wave approximation, the modified Jaynes-Cummings models are shown to be reduced to the conventional Jaynes-Cummings models with renormalized couplings between qubits and photon fields. A single-step Toffoli gate is shown to be realizable with tuning of the four characteristic oscillation periods that satisfy a commensurate condition. Possible values of the system parameters are estimated for a single-step Toffoli gate. Further, we discuss our single-step Toffoli-gate operation errors due to imperfections in the system parameters; numerical calculations show that a Toffoli gate with high fidelity can be obtained by adjusting pairs of the photon-qubit and qubit-qubit coupling strengths. In addition, a decoherence effect on the Toffoli-gate operation due to a thermal reservoir is discussed.

DOI
10.1103/physreva.85.032326

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Single-step implementations of multiqubit gates are generally believed to provide a simpler design, a faster operation, and a lower decoherence. For three coupled qubits interacting with a photon field, a realizable scheme for a single-step Toffoli gate is investigated. We find that the three-qubit system can be described by four effective modified Jaynes-Cummings models in the states of two control qubits. Within the rotating-wave approximation, the modified Jaynes-Cummings models are shown to be reduced to the conventional Jaynes-Cummings models with renormalized couplings between qubits and photon fields. A single-step Toffoli gate is shown to be realizable with tuning of the four characteristic oscillation periods that satisfy a commensurate condition. Possible values of the system parameters are estimated for a single-step Toffoli gate. Further, we discuss our single-step Toffoli-gate operation errors due to imperfections in the system parameters; numerical calculations show that a Toffoli gate with high fidelity can be obtained by adjusting pairs of the photon-qubit and qubit-qubit coupling strengths. In addition, a decoherence effect on the Toffoli-gate operation due to a thermal reservoir is discussed.
The family of nnn-bit Toffoli gates, with the two-bit Toffoli gate as the figurehead, are of great interest in quantum information as they can be used as universal gates and in quantum error correction, among other things. We present a single-step implementation of arbitrary nnn-bit Toffoli gates (up to a local change of basis), based on resonantly driving a single qubit that has a strong Ising coupling to nnn other qubits. The setup in the two-qubit case turns out to be identical to the universal Barenco gate. The gate time and error are, in theory, independent of the number of control qubits, scaling better than conventional circuit decompositions. We note that our assumptions, namely, strongly coupling n+1n+1n+1 qubits and a driving frequency that scales with nnn, may break down for large systems. Still, our protocol could enhance the capabilities of intermediate-scale quantum computers, and we discuss the prospects of implementing our protocol on trapped ions, Rydberg atoms, and superconducting circuits. Simulations of the latter platform show that the Toffoli gate with two control bits attains fidelities of above 0.98 even in the presence of decoherence. We also show how similar ideas can be used to make a series of controlled-not gates in a single step. We show how these can speed up the implementation of quantum error correcting codes and we simulate the encoding steps of the three-qubit bit-flip code and the seven-qubit Steane code.
Compared with the idea of universal quantum computation, a direct synthesis of a multiqubit logic gate can greatly improve the efficiency of quantum information processing tasks. Here we propose an efficient scheme to implement a three-qubit controlled-not (Toffoli) gate of neutral atoms based on unconventional Rydberg pumping. By adjusting the strengths of Rabi frequencies of driving fields, the Toffoli gate can be achieved within one step, which is also insensitive to the fluctuation of the Rydberg-Rydberg interaction. Considering different atom alignments, we can obtain a high-fidelity Toffoli gate at the same operation time ∼7 μs. In addition, our scheme can be further extended to the four-qubit case without altering the operating time.
Conditional multiqubit gates are a key component for elaborate quantum algorithms. In a recent work, Rasmussen et al. [Phys. Rev. A 101, 022308 (2020)] proposed an efficient single-step method for a prototypical multiqubit gate, a Toffoli gate, based on a combination of Ising interactions between control qubits and an appropriate driving field on a target qubit. Trapped ions are a natural platform to implement this method, since Ising interactions mediated by phonons have been demonstrated in increasingly large ion crystals. However, the simultaneous application of these interactions and the driving field required for the gate results in undesired entanglement between the qubits and the motion of the ions, reducing the gate fidelity. In this work, we propose a solution based on adiabatic switching of these phonon-mediated Ising interactions. We study the effects of imperfect ground-state cooling and use spin-echo techniques to undo unwanted phase accumulation in the achievable fidelities. For gates coupling to all axial modes of a linear crystal, we calculate high-fidelity (>99(>99%)(>99 NNN-qubit rotations with N=3−−7N=3--7N=3−−7 ions cooled to their ground state of motion and a gate time below 1 ms. Finally, we study the effect of laser intensity fluctuations and find that the proposed gate requires intensity stabilization with subpercentage noise levels. The high fidelities obtained also for large crystals could make the gate competitive with gate-decomposed, multistep variants of the NNN-qubit Toffoli gate, at the expense of requiring ground-state cooling of the ion crystal.
The Toffoli gate (controlled-controlled-NOT gate) is one typical three-qubit gate, it plus a Hadamard gate form a universal set of gates in quantum computation. We present an efficient method to implement the Toffoli gate using an array of coupled cavities with one three-level atom in each cavity. The large detuning between atoms and classical (quantum) fields plays an important role and the gate is implemented in one-step. The quantum information is encoded into the low-lying states of identical atoms and it is convenient to address qubit individually. Based on the Markovian master equation, it is shown that the scheme to implement the Toffoli gate is robust against the decoherence.
We propose a protocol for implementing Barenco-type multi-qubit controlled gates using short driven spin chains. Starting from an Ising interaction with a transverse drive on the last spin, we construct an effective two-qubit Hamiltonian whose time evolution implements the Barenco gate V2(φ,ω,φ)V_2(\varphi,ω,φ)V2​(φ,ω,φ) and, in particular, a CNOT gate. We then embed this construction into a three-qubit XXZXXZXXZ chain to realize the three-qubit Barenco gate V3(φ,ω,φ)V_3(\varphi,ω,φ)V3​(φ,ω,φ), which includes the Toffoli gate as a special case. The derivation is fully analytical: we perform a sequence of unitary transformations, identify decoupled subspaces, and apply a rotating-wave approximation to obtain simple effective Hamiltonians. We derive explicit conditions on the coupling strengths and driving parameters, provide closed-form expressions for the time-evolution operators in each relevant subspace, and characterize the quality of the implementation using the operator fidelity. Numerical simulations show that the protocol achieves high fidelities over broad parameter ranges, demonstrating its robustness and suitability for quantum information processing in spin-chain platforms.
We propose a one-step scheme to implement a multiqubit controlled phase gate with one qubit simultaneously controlling multiple qubits with three-level atoms at distant nodes in coupled cavity arrays. Selective qubit-qubit couplings are achieved by adiabatically eliminating the atomic excited states and photonic states, and the required phase shifts between the control qubit and any target qubit can be realized through suitable choices of the parameters of the external fields. Moreover, the effective model is robust against decoherence because neither the atoms nor the field modes are excited during the gate operation, leading to a useful step toward scalable quantum computing networks.